#### marcus

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Larsson's POV puzzles me and maybe someone here can help me by explaining

---post 4 of his diffeo thread---

Helling-Policastro showed that canonical quantization a la LQG is

strange. However, it might be worth pointing out that string theory

admits no canonical quantization at all in more than two dimensions.

Everything is done using path integrals. Usually one justifies formal

manipulations with path integrals by referring to the well-defined

Hamiltonian formalism. This seems somewhat dubious if no Hamiltonian

formalism exists.

This puts some perspective on LQG's achievements. It is not obvious

to me that strange canonical quantization is so much worse than no

canonical quantization at all.

The secret reason why canonical quantization of diff-invariant

theories in more than 2D fails is that the relevant diffeomorphism

group anomaly is little known. The diffeomorphism generators should

be represented by unitary operators on a conventional Hilbert space,

and all non-trivial such representations are anomalous. Since neither

the string theory nor LQG camps care about these anomalies in 4D,

they cannot do canonical quantization.

---end quote---

He seems to have a special idea of what is canonical quantization, requiring that there be a unitary rep of the diffeo group. then he shows why Loop (or String, whatever) cannot have a unitary rep of the diffeo group.

then he concludes that they cannot have (his idea of) canonical quantization.

but when I read mainstream LQG people they do not say they have a unitary rep of the diffeos, and they have been saying for over 10 years that the LQG approach is canonical quantization of GR. they obviously MEAN something different from Larsson when they say canonical. His requirement for what it means is more stringent, apparently.

Larsson applies his demands even-handedly, I must say. Here is a recent post at NEW. This is yesterday (12 October) on the Witten thread

---quote from Larsson---

response to:

__If you were Witten, what would you do to "fix up" string theory as it's known today (besides fixing up diffeomorphism anomalies)?__

What would convince you to change your mind and be in support of string theory?

What would convince you to change your mind and be in support of string theory?

In the unlikely event that string theory acquired massive experimental support, I guess that I would have to believe in it. But the present situation is rather the opposite.

The construction of a quantum theory with some prescribed symmetries is, from my perspective, the same thing a constructing the representation theory of the group of symmetries. There is really a 1-1 correspondence:

1. Given a quantum theory, its symmetry group acts by a unitary representation on the Hilbert space.

2. Given a unitary representation of some group, the Hilbert space on which it acts is the Hilbert space of some quantum theory.

In particular, the Hilbert spaces of the fully interacting gauge-invariant or diff-invariant theories carry unitary representation of the groups of gauge transformations and diffeomorphisms. Perhaps one should factor out gauge symmetries, although I don't see why - it is definitely not necessary for consistency (unitarity). But this is really irrelevant for the argument. The anomalies must be there at least before factoring them out, so if you cannot write down the anomalies in the first place, you lose.

I am pretty sure that there is no way to fix string theory. The representations look the way they do, and their Hilbert spaces look rather like fixed versions of field theory. I don't see any way to "fix" SU(2) to allow for unitary spin-1/4 representations either.

I don't have a clue what I would do if I were Witten, and I don't really care. It's not my problem.

Posted by: Thomas Larsson at October 12, 2004 12:20 PM

---end quote---